large N limit (Rev #4) in nLab

Contents

Idea

In theoretical physics, one often considers gauge theory models whose symmetries are groups of large matrices, e.g. U(N)U(N) or O(N)O(N). The limit of such theories for N→∞N\to \infty has often remarkable properties and string-like features.

This limit is sometimes appearing in its own right, but sometimes it is just considered as an approximation for a system with fixed finite NN. One of the features is that in the large N limit is that non-planar Feynman diagrams loose their importance and that the correlation functions satisfy certain decoupling/factorization rule. The behaviour is studied in terms of expansion in 1/N1/N whose square has a simialr role to Planck constant in semiclassical approximation limit of quantum mechanics.

• AdS-CFT duality

References

• A. A. Migdal, Loop equations and 1/N expansion, Physics Reports, 102 (4), 199-290 (1983) doi
• Laurence G. Yaffe, Large N limits as classical mechanics, Rev. Mod. Phys. 54, 407–435 (1982) pdf
• G ‘t Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72, 3 (1974), 461-473, doi
• G. ‘t Hooft, Large N, workshop lecture, hep-th/0204069
• S. Coleman, 1/N, in Aspects of Symmetry, Cambridge University Press 1985.
• A. V. Manohar, Large N QCD, L:es Houches Lecture 2004, pdf
• A. Jevicki, Instantons and the 1/N1/N expansion in nonlinear σ\sigma models, Phys. Rev. D 20, 3331–3335 (1979) pdf
• Juan M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv.Theor.Math.Phys.2:231-252, 1998 hep-th/9711200; Wilson loops in large N field theories, hep-th/9803002.
• E. Brézin, S.R. Wadia, eds. The Large N Expansion in Quantum Field Theory and Statistical Physics, a book collection of reprinted historical articles, gBooks
• Yuri Makeenko, Methods of contemporary gauge theory, Cambridge Monographs on Math. Physics, gBooks
• M. Bershadsky, Z. Kakushadze, C. Vafa, String expansion as large N expansion of gauge theories, Nucl.Phys. B523 (1998) 59-72hep-th/9803076, doi
• G.T. Horowitz, H. Ooguri, Spectrum of large N gauge theory from supergravity, hep-th/9802116
• Semyon Klevtsov, Random normal matrices, Bergman kernel and projective embeddings, arxiv/1309.7333
• wikipedia 1/N expansion